NoteNextra-origin/content/Math401/Math401_N1.md

Node 1

all the materials are recovered after the end of the course. I cannot split my mind away from those materials.

Recap on familiar ideas

Group

A group is a set G with a binary operation \cdot that satisfies the following properties:

1. Closure: For all a, b \in G, the result of the operation a \cdot b is also in G.
2. Associativity: For all a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c).
3. Identity: There exists an element e \in G such that for all a \in G, e \cdot a = a \cdot e = a.
4. Inverses: For each a \in G, there exists an element b \in G such that a \cdot b = b \cdot a = e.

Ring

A ring is a set R with two binary operations, addition and multiplication, that satisfies the following properties:

1. Additive Closure: For all a, b \in R, the result of the addition a + b is also in R.
2. Additive Associativity: For all a, b, c \in R, (a + b) + c = a + (b + c).
3. Additive Identity: There exists an element 0 \in R such that for all a \in R, 0 + a = a + 0 = a.
4. Additive Inverses: For each a \in R, there exists an element -a \in R such that a + (-a) = (-a) + a = 0.
5. Multiplicative Closure: For all a, b \in R, the result of the multiplication a \cdot b is also in R.
6. Multiplicative Associativity: For all a, b, c \in R, (a \cdot b) \cdot c = a \cdot (b \cdot c).

Others not shown since you don't need too much.

Symmetric Group

Definition

The symmetric group S_n is the group of all permutations of n elements. Or in other words, the group of all bijections from a set of n elements to itself.

Example:

e=1,2,3\\
(12)=2,1,3\\
(13)=3,2,1\\
(23)=1,3,2\\
(123)=3,1,2\\
(132)=2,3,1

(12) means that 1\to 2, 2\to 1, 3\to 3 we follows the cyclic order for the elements in the set.

S_3 = \{e, (12), (13), (23), (123), (132)\}

The multiplication table of S_3 is:

Element	e	(12)	(13)	(23)	(123)	(132)
e	e	(12)	(13)	(23)	(123)	(132)
(12)	(12)	e	(123)	(13)	(23)	(132)
(13)	(13)	(132)	e	(12)	(23)	(123)
(23)	(23)	(123)	(132)	e	(12)	(13)
(123)	(123)	(13)	(23)	(132)	e	(12)
(132)	(132)	(23)	(12)	(123)	(13)	e

Functions defined on S_n

Symmetric Generating Set